Connectivity of generating graphs of nilpotent groups

TL;DR

This paper investigates the connectivity and Hamiltonian properties of generating graphs of finite nilpotent groups, revealing they are maximally connected and Hamiltonian after removing isolated vertices.

Contribution

It establishes that generating graphs of finite nilpotent groups are maximally connected and Hamiltonian, advancing understanding of their combinatorial structure.

Findings

Generated graphs are maximally connected.

Graphs are Hamiltonian for groups of order at least 3.

Isolated vertices removal yields maximally connected graphs.

Abstract

Let $G$ be $2$-generated group. The generating graph of $\Gamma(G)$ is the graph whose vertices are the elements of $G$ and where two vertices $g$ and $h$ are adjacent if $G=\langle g,h\rangle$. This graph encodes the combinatorial structure of the distribution of generating pairs across $G$. In this paper we study several natural graph theoretic properties related to the connectedness of $\Gamma(G)$ in the case where $G$ is a finite nilpotent group. For example, we prove that if $G$ is nilpotent, then the graph obtained from $\Gamma(G)$ by removing its isolated vertices is maximally connected and, if $|G| \geq 3$, also Hamiltonian. We pose several questions.